THE THEORY OF INTERSTELLAR TRADE
PAUL KRUGMAN∗
This article extends interplanetary trade theory to an interstellar tting.It is chiefly concerned with the following question:how should interest charges on goods in transit be computed when the goods travel at clo to the speed of light?This is a problem becau the time taken in transit will appear less to an obrver traveling with the goods than to a stationary obrver.A solution is derived from economic theory,and two uless but true theorems are proved.(JEL F10,F30)
I.INTRODUCTION
Many critics of conventional economics have argued,with considerable justification,that the assumptions underlying neoclassical theory bear little remblance to the world we know.The critics have,however,been too quick to asrt that this shows that mainstream economics can never be of any u.Recent progress in the tech-nology of space travel as well as the prospects of the u of space for energy production and colonization(O’Neill1976)make this asrtion doubtful;for they rai the distinct possibility that we may eventually discover or construct a world to which orthodox economic theory applies.It is obvious,then,that economists have a special interest in understanding
and,indeed, in promoting the development of an interstellar economy.One may even hope that formula-tion of adequate theories of interstellar economic relations will help accelerate the emergence of such relations.Is it too much to suggest that current work might prove as influential in this development as the work of Adam Smith was in the initial ttlement of Massachutts and Virginia?
This article reprents one small step for an economist in the direction of a theory of inter-stellar trade.It goes directly to the problem of trade over stellar distances,leaving aside the analysis of trade within the Solar System.Inter-planetary trade,while of considerable empir-ical interest(Frankel1975),rais no major *This rearch was supported by a grant from the Committee to Re-Elect William Proxmire.This article is adapted with minor changes from a manuscript written in July1978.
Krugman:Professor,Woodrow Wilson School,Princeton University,Princeton,NJ08544-1013.Phone609-258-1548,Fax609-258-0019,E-mail pkrugman@princeton.
edu
theoretical problems since it can be treated in the same framework as interregional and inter-nationa
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l trade.Among the authors who have not pointed this out are Ohlin(1933)and Samuelson (1947).Interstellar trade,by contrast,involves wholly novel considerations.The most impor-tant of the are the problem of evaluating capi-tal costs on goods in transit when the time taken to ship them depends on the obrver’s refer-ence frame;and the proper modeling of arbitrage in interstellar capital markets where—or when (which comes to the same thing)—simultaneity ceas to have an unambiguous meaning.
The complications make the theory of interstellar trade appear atfirst quite alien to our usual trade models;presumably,it ems equally human to alien trade theorists.But the basic principles of maximization and opportu-nity cost will be en to give clear answers to the questions.I do not pretend to develop here
a theory that is universally valid,but it may at
least have some galactic relevance.
The remainder of this article is,will be,or has been,depending on the reader’s inertial frame, divided into three ctions.Section II develops the basic Einsteinian framework of the analysis.
In Section III,this framework is ud to ana-lyze interstellar trade in goods.Section IV then considers the role of interstellar capital move-ments.It should be noted that,while the subject of this article is silly,the analysis actually does make n.This article,then,is a rious anal-ysis of a ridiculous subject,which is of cour the opposite of what is usual in economics.
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II.FUNDAMENTAL CONSIDERATIONS There are two major features distinguishing interstellar trade from the interplanetary trade we are accustomed to.Thefirst is that the time
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Economic Inquiry doi:10.1111/j.1465-7295.2009.00225.x
(ISSN0095-2583)Online Early publication March7,2010
V ol.48,No.4,October2010,1119–1123©2010Western Economic Association International
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1120ECONOMIC INQUIRY
spent in transit will be very great since travel must occur at less than light speed;round trips of veral hundred years appear likely.The cond is that,if interstellar trade is to be at all practical, the spaceships that conduct it must move at speeds that are reasonable fractions of the speed of light.
Becau interstellar trade will take so long, any decision to launch a cargo will necessar-ily be a very long-term investment project and would hardly be conceivable unless there are very extensive futures markets.I will assume, then,that future futures markets are,well,futur-istic in their development.In fact,I will assume that investors,human or otherwi,are able to make perfect forecasts of prices over indefinite periods.
The cond feature of interstellar transactions cannot be so easily dealt with(physicists are not as tolerant as economists of the practice of assuming difficulties away).If trading space vesls move at high velocities,we can no longer have an unambiguous measure of the time taken in transit.The time taken by the spacecraft to make a round trip will appear less to an obrver on the craft than to one remaining on Earth.Since an interstellar voyage is an investment project that must have a positive prent value,there is obviously a problem in deciding which transit time to u in the prent value calculation.
This is an inertial problem—which becomes a weighty problem in a gravitationalfield—requiring an economic analysis,provided in the next ction.In this ction,I develop the necessary physical concepts,illustrated in Figure1.Consider trade between two planets, Earth and Trantor.I assume that the two planets may be regarded as being in the same inertial frame.Then their world lines in space-t
ime can be reprented by two parallel lines,shown as EE E and TT T in thefigure.Several types of contact between the two planets are also shown.The line ET is the world line of an electromagnetic signal—say,a rerun of Star Trek—nt from Earth to Trantor.If time is measured in years and space in light years,ET will have a45◦slope.The line E T is the world line of a spaceship moving with uniform velocity from Earth to Trantor.It must be steeper than ET becau the spaceship’s speed must be less than that of light.Finally,E T shows a spaceship path,which is more likely in practice:it involves initial acceleration,followed by deceleration.
FIGURE1
World Lines in Space-Time between Earth and
Trantor
The problem of time dilation must now be considered.It will suffice here to consider the ca of a spaceship with uniform velocity.It is then well known—e,for example,Lawden (1962)—that if the voyage from Earth to Tran-tor appears to take n years to obrvers in the Earth-Trantor inertial reference frame,it will appear to take n years aboard the spaceship, where:
n=n
1−
v2
c2
,
(1)
where v is the spacecraft’s velocity and c the speed of light.This can easily be demonstrated by reprenting the voyage in Minkowski space-time,that is,with a real space axis and an imag-inary time axis.The ship’s velocity can then be reprented by a rotation of the axes;the rotation of the time axis is shown in Figure2.(Readers whofind Figure2puzzling should recall that a diagram of an imaginary axis must,of cour, itlf be imaginary.)
KRUGMAN:THEORY OF INTERSTELLAR TRADE1121
FIGURE2
Rotation of the Imaginary Time Axis in
Minkowski Space-Time
To conclude this ction,we should say something about the assumption that the trading planets lie in the same inertial frame.This will turn out to be a uful simplification,permitting us to limit ourlves to consideration of special relativity.It is also a reasonable approximation for tho planets with which we are likely to trade.Readers may,however,wish to u general relativity to extend the analysis to trade between planets with large relative motion.This extension is left as an exerci for i
nterested readers becau the author does not understand the theory of general relativity,and therefore cannot do it himlf.
III.INTERSTELLAR TRADE IN GOODS We are now prepared to begin the economic analysis.Let us start with some notation.Let p E,p T=price of Terran,Trantorian goods on Earth
p∗
E
,p∗
T
=price of Terran,Trantorian goods on Trantor
r,r∗=interest rates on Earth,Trantor
N=number of years taken to travel from Earth to Trantor(or vice versa),as measured by an obrver in the Earth-Trantor inertial frame.
All the quantities except N should,of cour,be defined at a point in time;except where specified,however,I will make the sim-plifying assumption that the quantities are in fact constant over time.
evolutionaryNow let us begin by considering the simplest kind of interstellar transaction,one which will reveal the problems of analysis and also give us the key to their solution.Suppo a Trantorian merchant decides to consider trading with Earth. Assume,provisionally,that interest rates are the same on both planets.(This assumption will be justified in the next ction.)Then,it(the merchant)may have in its mind(or equivalent organ)a ries of transactions of the following kind.It will make an initial expenditure of
c+q∗
T
p∗
T
,where c is the cost of outfitting a ship and q∗T is the quantity of Trantorian goods shipped.When the
ship reaches Earth,the goods will be exchanged for a quantity of Earth goods;given the notation already developed,this quantity will be q∗E=q∗T p T p
E
.Finally,on return, the goods will be sold at the price p∗E,yielding revenue q
∗
T
p T p∗E
p E
.
Is this transaction profitable?A merchant staying home on Trantor will ask whether the prent value of the revenue exceeds the initial cost;since the trip takes2N years from the point of view of a stationary obrver,the test criterion is:
q∗
T
p T p∗
E
p E
≥(c+q∗T p∗T)(1+r∗)2N. (2)
But suppo the merchant had traveled with its cargo?The trip would then,from its point of view,have taken only2N
1−v2
c2
years, suggesting an alternative criterion of acceptance,
网络销售培训q∗
T
p T p∗
E
p E
≥(c+q∗T p∗T)(1+r∗)2N
1−
v2新东方西安分校
c2 (2 )
The criteria cannot both be right.Which,then, is correct?
The answer may be obtained by consider-ing the justification for prent value calcula-tions.A prent value calculation makes n becau it takes account of opportunity cost:an investor might,instead of undertaking a project,
1122ECONOMIC INQUIRY have bought a bond.In this ca,the merchant
might have bought a bond on Trantor and let it
mature instead of nding a cargo to Earth.The
value of the bond on the ship’s return does not
depend on the time elapd on board the ship
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itlf.So Equation(2),not Equation(2 ),is the
proper criterion.We have thus demonstrated the following.
First Fundamental Theorem of Inter-
stellar Trade:When trade takes place
between two planets in a common inertial
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frame,the interest costs on goods in transit
should be calculated using time measured
by clocks in the common frame and not by
clocks in the frames of trading spacecraft.
At this point,it is unlikely that the reader will
rai the following objection.Suppo that the merchant,instead of making a round trip,were
to travel with its cargo and ttle down on Earth
as well,not man,but say a rich being.
Would the argument still be valid?
We can most easily e that the argument is
still valid if we consider a special ca.Suppo
that the transportation costs other than interest
on goods in transit are negligible;and suppo
further that the interstellar shipping industry is competitive,so that profits are driven to zero.
Then,if Equation(2)is a correct criterion we
have the relationship:
p∗E p∗T =
p E
p T
(1+r∗)2N.
(3)
Thus,relative goods prices will not be equalized; rather,there will be a wedge driven between relative prices on Earth and on Trantor.
Now,within this special ca,consider the position of a Trantorian planning to migrate to Earth.It could purcha a cargo on Trantor and ll it on Earth.Alternatively,though,it could buy a bond on Trantor and,on reaching Earth, ll its claim to a Terran planning to travel in the opposite direction.Becau of this alternative possibility,the fact that the merchant itlf does not plan to make a round trip is inesntial since what the Terran will be willing to pay for the claim will reflect the extent to which its value will have grown on Trantor when the Terran arrives.A one credit(Trantorian)bond,bought by a merchant just about to migrate,will have grown in value to CrT.(1+r∗)2N by the time a migrant in the other direction can arrive to claim it.Such a migrant would have the choice of buying the bond or carrying Earth goods with him,so arbitrage will mean that the price of the claim on Earth will be CrE.(1+r∗)2N
p E
p∗E
rbs
.
But one credit(Trantorian)worth of cargo shipped from Trantor to Earth will ll for CrE.
p T
p∗T
,which by Equation(3)is equal to CrE.(1+r∗)2N
p E
p∗E
.So the Trantorian mer-chant will be indifferent between shipping goods and buying a bond.This shows that the First Fundamental Theorem of Interstellar Trade remains valid,even if no spacecraft or individ-uals make round trips.All that is necessary is that there be two-way trade,with somebody or something going in each direction.
This proof has been for a special ca;but the proposition is in fact relatively general.(The reader must,of cour,be careful not to confu relative generality with general relativity.)A proof of the First Fundamental Theorem in the prence of transportation costs may be found in an unwritten working paper by the author (Krugman1987).
IV.INTERSTELLAR CAPITAL MOVEMENTS
Alert readers will have noticed that the analy-sis of interstellar trade in goods already involves some discussion of ast markets,both becau interstellar transportation costs depend on inter-est rates and becau the validity of the First Fundamental Theorem depends on arbitrage through interspecies transactions in curities. Further,the results of the last ction depended on the assumption of equal interest rates on the two planets.In this ction,we will examine the effects of interstellar capital movements.In particular,we want to know whether interstellar arbitrage will in fact equalize interest rates.
One might atfirst doubt this.Arbitrage is possible internationally becau an investor can choo between holding his wealth in different countries for the next,say,30days simply by calling up his broker and instructing him. In interstellar trade,things are not so simple. Even if we leave on one side the problem that nonhuman brokers may not have ears, let alone telephones,there is the problem that simultaneous arbitrage is not possible.Messages must travel at light speed;goods more slowly still.We have already en that this means that relative goods prices will vary from planet to planet,even if there are no transportation costs in the usual n.Will not interest rates differ as well?
KRUGMAN:THEORY OF INTERSTELLAR TRADE1123 Perhaps surprisingly,the answer is no.It will
suffice to consider a particular example of an
interstellar capital transaction.Suppo that,as
in the last ction,interest costs on goods in
transit are the only transportation costs.Then,
Equation(3)will hold for relative prices.Now
let a Trantorian resident carry out the follow-
ing t of transactions:(1)it ships goods to
Earth;(2)it then invests the proceeds from ll-高一必修一英语单词
ing the goods in Terran bonds for K years;
(3)it then buys Terran goods and ships them
to Trantor.The return on this t of transac-
tions,viewed as an investment,must be the same
as the return on holding bonds for the same
period,that is,2N+K years.This gives us the
condition:
(1+r∗)2N+K=
p∗
E
p∗
T
p T
p E
(1+r)K.
(4)
But if we u relationship(3),this reduces to r=r∗.We have thus arrived at the result that interest rates will be equalized.
Second Fundamental Theorem of Inter-
stellar Trade:If ntient beings may hold
asts on two planets in the same inertial
frame,competition will equalize the inter-
est rates on the two planets.
Combining the two theorems developed in this article,it will be en that we have the foundation for a coherent theory of interstellar trade between planets in the same inertial frame.Interstellar trading voyages can be regarded as investment projects,to be evaluated at an interest rate that will be com
mon to the planets. From this point,the effects of trade on factor prices,income distribution,and welfare can be traced out using the conventional tools of general equilibrium analysis.The picture of the world—or,rather,of the univer—which emerges is not a lunatic vision;stellar,maybe, but not lunatic.
Is space the Final Frontier of economics? Certainly this is only afirst probe of the sub-ject,but the possibilities are surely limitless. (In curved space-time,of cour,this does not prevent the possibilities from beingfinite as well!)I have not even touched on the fasci-nating possibilities of interstellarfinance,where spot and forward exchange markets will have to be supplemented by conditional prent markets. Tho of us working in thisfield are still a small band,but we know that the Force is with us.
REFERENCES
Frankel,J.“Is There Trade with Other Planets?”Washing-ton,DC:International Monetary Fund,1975. Krugman,P.“Theory Capital and Travel Light-than-Faster.”New Haven,CT:Yale University,1987. Lawden, D. F.An Introduction to Tensor Calculus and Relativity.New York:Wiley,1962.
Ohlin,B.Interregional and International Trade.Cambridge, MA:Harvard University Press,1933.
O’Neill,G.The High Frontier.New York:William Morrow, 1976.
Samuelson,P.Foundations of Economic Analysis.
Cambridge,MA:Harvard University Press,1947.
